Problem 5
Solve the differential equation y''' = sin x by successive…
Solve the differential equation $y''' = \sin x$ by successive integration.
Explanation of Method
The right-hand side depends only on $x$, so integrate three times, introducing a new arbitrary constant at each step.
Step 1: First integration
\[y'' = \int \sin x\,dx = -\cos x + C_1.\]
Step 2: Second integration
\[y' = \int(-\cos x + C_1)\,dx = -\sin x + C_1 x + C_2.\]
Step 3: Third integration
\[y = \int(-\sin x + C_1 x + C_2)\,dx = \cos x + \frac{C_1}{2}x^2 + C_2 x + C_3.\]
Step 4: General solution
Absorbing $C_1/2$ into a renamed constant $C_1$,
\[y = \cos x + C_1 x^2 + C_2 x + C_3,\] where $C_1$, $C_2$, and $C_3$ are arbitrary constants.
Step 5: Verification
Differentiating three times: $y' = -\sin x + 2C_1 x + C_2$, $y'' = -\cos x + 2C_1$, $y''' = \sin x$.
Result
Final answer:
\[\boxed{\,y = \cos x + C_1 x^2 + C_2 x + C_3\,}\]